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On a theorem of Mahler


Authors: N. K. Meher, K. Senthil Kumar and R. Thangadurai
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 11K16
DOI: https://doi.org/10.1090/proc/13616
Published electronically: May 26, 2017
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Abstract: Let $ b\geq 2$ be an integer and $ \alpha $ a non-zero real number written in base $ b$. In 1973, Mahler proved the following result: Let $ \alpha $ be an irrational number written in base $ b$ and let $ n\geq 1$ be a given integer. Let $ B = b_0b_1\ldots b_{n-1}$ be a given block of digits in base $ b$ of length $ n$. Then, there exists an integer $ X$ with $ 1\leq X < b^{2n+1}$ such that $ B$ occurs infinitely often in the base $ b$ representation of the fractional part of $ X\alpha $. In this short note, we deal with some conditional quantitative version of this result.


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Additional Information

N. K. Meher
Affiliation: Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad, 211019, India
Email: nabinmeher@hri.res.in

K. Senthil Kumar
Affiliation: Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam, Chennai, 603103, India
Address at time of publication: National Institute of Science Education and Research, HBNI, P.O. Jatni, Khurda 752050, Odisha, India
Email: senthil@niser.ac.in

R. Thangadurai
Affiliation: Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad, 211019, India
Email: thanga@hri.res.in

DOI: https://doi.org/10.1090/proc/13616
Keywords: Algebraic numbers, Borel's conjecture, Diophantine inequalities, digits
Received by editor(s): September 25, 2016
Received by editor(s) in revised form: November 1, 2016, and November 28, 2016
Published electronically: May 26, 2017
Dedicated: Dedicated to Michel Waldschmidt on his 70th birthday
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society